arXiv research

Study knot Floer cohomology using hyperbolic metrics and wrapped Fukaya categories.

We define a torus algebra for Heegaard Floer homology.

Researchers create a Fukaya category for knot complements using specific metrics.

Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.

Paper establishes an isomorphism between Fukaya category and bordered knot Floer homology.

Proves Arnol'd's chord conjecture for conormal bundles.

Study shows links with unbounded annular Khovanov gradings but bounded Floer gradings.

Study volume growth in Milnor fibers using real Lagrangians.

We prove that the wrapped Fukaya category of any $2n$-dimensional Weinstein manifold (or, more generally, Weinstein sector) $W$ is generated by the unstable manifolds of the index $n$ critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer cohomology and t…

The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and Lagrangian correspondences. More specifically, we give a description of the algebra A(F) which appears in the work of Lipshit…

We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic c…

A mathematical isomorphism connects Floer homology to DAHA representations.

The purpose of this thesis is to define a "local" version of Ozsváth and Szabó's Heegaard Floer homology $\operatorname{\widehat{HFL}}$ for links in the 3-dimensional sphere, i.e. a Heegaard Floer homology $\operatorname{\widehat{HFT}}$ for tangles in the closed 3-ball. After studying basic properties of $\operatorname…

Study Morse theory on loop spaces and Hecke algebras.

New connection found between shape reconstruction methods and persistent homology.

The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.

We introduce a wrapped Gaussian for SPD matrices, enhancing data analysis.

Multi-output is essential in machine learning that it might suffer from nonconforming residual distributions, i.e., the multi-output residual distributions are not conforming to the expected distribution. In this paper, we propose "Wrapped Loss Function" to wrap the original loss function to alleviate the problem. This…

New examples of Legendrian links with infinitely many fillings.

PAC-Wrap provides provable guarantees for semi-supervised anomaly detection.

Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they …

Formula for satellite operators using knot Floer homology.

The paper introduces exponential-wrapped distributions on symmetric spaces for better data modeling.

Link Floer homology is split into snake complexes and local systems.

Proves a conjecture for annular links using homology classes.

New physics-based method calculates Reidemeister torsions for hyperbolic 3-manifolds.

Algorithm computes knot Floer complex for knots of thickness one.

This is the second of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an a…

The paper classifies knot Floer complexes of low width, simplifying knot bases.

Geometric interpretation of tangle invariants using immersed curves.

Conjecturally, there are only finitely many Heegaard Floer L-space knots in $S^3$ of a given genus. We examine this conjecture for twist families of knots $\{K_n\}$ obtained by twisting a knot $K$ in $S^3$ along an unknot $c$ in terms of the linking number $ω$ between $K$ and $c$. We establish the conjecture in case of…

Study algebraic curves in C^2 using Floer theory.

We analyze the effect of adding, removing, and moving basepoints on link Floer homology. We prove that adding or removing basepoints via a procedure called quasi-stabilization is a natural operation on a certain version of link Floer homology, which we call $CFL_{UV}^\infty$. We consider the effect on the full link Flo…

New method for Lagrangian Floer homology groups using flow trees.

Given a finite set of points in $\mathbb R^n$ and a radius parameter, we study the Čech, Delaunay-Čech, Delaunay (or Alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four…

New method computes knot Floer homology for satellite knots.

New method connects knot Floer homology with bordered Floer homology.

Develops Floer theory for 3-manifold covers using equivariant structures.

This is the third of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an au…

We define and prove properties of link lattice complexes for plumbed links.

A formula for bordered Floer homology of concordances and satellites

We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this coro…

Study on singularities of Lagrangian immersions with applications in Floer theory.

We study the behavior of $\mathrm{Pin}(2)$-monopole Floer homology under connected sums. After constructing a (partially defined) $\mathcal{A}_{\infty}$-module structure on the $\mathrm{Pin}(2)$-monopole Floer chain complex of a three manifold (in the spirit of Baldwin and Bloom's monopole category), we identify up to …

For any three-manifold presented as surgery on a framed link (L,Λ) in an integral homology sphere, Manolescu and Ozsváth construct a hypercube of chain complexes whose homology calculates the Heegaard Floer homology of Λ-framed surgery on Y. This carries a natural filtration that exists on any hypercube of chain comple…

Proves rigidity of sphere metrics with subsets removed.

Heegaard Floer homology study confirms composition maps match up to homotopy.

Knot Floer homology is reinterpreted as immersed curves.